Nucleation in the two-dimensional Ising model

Abstract

The aim of this thesis is to study nucleation both numerically and analytically. The approach followed is to start with very simple models. In chapter 2 we study the Ising model without an external magnetic field. This system does not feature nucleation, but at low temperatures it jumps back and forth between a state in which most spins are up, to one in which most spins are down. The dominant pathway at low temperatures consists of the formation of a single pair of closed interfaces in the shorter periodic direction, which perform a relative diffusive motion around the longer periodic direction and annihilate after meeting each other through the periodic boundary.

In chapter 3 we study the Ising model with an external field on a square lattice. Initially the system is in a metastable state, with most of the spins anti-aligned with the external field. It will stay in this metastable state for an extended period of time, but eventually one of the small clusters of aligned spins that arise due to fluctuations, will grow beyond the critical cluster size, and take over the whole system. After this, most of the spins are aligned with the external field, and the system is in its stable state.

In chapter 4 the same model is studied. The effective rates of growth and shrinkage of clusters are now studied in detail. The mass of the nucleating cluster is followed in time, and mapped to a random walker undergoing drift and diffusion. The latter is described by the Fokker-Planck equation, in which in our case the drift and diffusion coefficients depend on cluster size.

The method developed in chapter 4 is then applied to a completely different phenomenon: the fluctuations of the geomagnetic dipole. Instead of to the time-evolution of the size of the nucleus, we apply our method of analysis, based on the Fokker-Planck equation, to the time-evolution of the strength of the geomagnetic dipole, which has been measured accurately over the last 800,000 years from fossile records. This is described in chapter 5.